Part of K C Cole’s teaser for tomorrow’s Categorically Not! – Inside Out said:

Sometimes the results are surprising: circles in the plane canâ€™t be turned inside out, but spheres in 3-dimensional space can be.

This is all the license I need to show you* this wonderful 21 minute video showing exactly that (and explaining some rather beautiful mathematics along the way):

Direct link to the Google Video (for larger viewing) here.

Enjoy!

-cvj

Is there a direct link between how we use consciousness and mathematics to present “consistency in our sociological thinking?”

So if we used a circle and a sphere “more correctly” we can expand the philosophy we are developing according to the inhernet nature of that math? Would you want too?

[…] Mathematician Danny Calegari gave a rather nice description of some of the things you might stumble upon in asking questions about the difference between inside and outside in a mathematical context, motivating rather nicely some aspects of the study of topology. He showed some simple diagrams, arriving very skillfully at the idea of a topological “invariant”. He then went on to give some examples of how you can change one space into another space (if they have the same invariant), and how you can’t (if they don’t have the same invariant). (The clickable images left shows how to turn a certain squiggle into a straight line as a result of them having the same invariant – you can get rid of the kinks without creasing the curve.) That you can’t turn a circle inside out in two dimensions (without such creases) while you can turn a two-sphere inside out in three dimensions was his final example. I recently showed you a video describing these last two examples, by the way. A major point of resonance with some of the audience (it was remarked) was that in his explanations, various concepts involved adopting different points of view to build up the whole picture – for example, the winding number of a curve from an observer at a given point can be thought of in terms of how many times your head has to turn around in watching a driver doing a circuit of the curve. People seemed to like that a lot, interestingly – getting something considered so mathematically precise from something that seems so everyday. I was rather pleased that people could see that it is not so big a leap from the everyday to interesting mathematics. […]

[…] Ever wanted to know how to turn your head inside out? Watch this! Without doubt the best math’s explanation video I’ve ever seen. [via Clifford] […]

Hi, Professor. Johnson

This is John Chae from phys408b last semester.

I was looking around your website and happened to wonder into this site. Actually, the reason was the name of this website. )

This video clip was REALLY amazing. I never thought about it. Of course.

I was amazed by the idea and its lower dimension analogy to a circle. I will visit

whenever I have time from now on.

bye

John

Hi John!

I hope all is well.

Thanks for stopping by! Come back soon!

-cvj